8.2.2.1 Fick’s First Law
Fick discovered by direct observation that the magnitude of the mass flux is
proportional to the magnitude of the concentration gradient in isotropic continuum
is given by.
J ¼ ÀD∇C
ð8:1Þ
where:
J—is the mass flux, which is defined as a vector quantity, with units:
mol m
À2 s
À1 .
D—is diffusivity, with unit’s m
2 s
À1 ; it is a function of D ¼ D 0 exp À
Q d
RT
À
Á
where
D 0 is temperature-independent constant, Q d is activation energy for diffusion, R is
the gas constant (8.31 J/mol K), and T is temperature in Kelvin.
C—is the concentration of the matter in question.
Combining the mass conservation equation with Fick’s first law yields the Fick’s
second law;
8.2.2.2 Fick’s Second Law: (for Nonsteady-State Diffusion)
Fick’s second law can be given in its most elementary form as.
∂C
∂t
¼ D∇
2 C
ð8:2Þ
Until the twentieth century, diffusion in materials was addressed at microscopic
scale only. Einstein showed the consistency of random motion of macroscopic
particles in the presence of molecules. Einstein derived a relationship for the chaotic
motion of small particles as.
v D ¼
DF
k B T
ð8:3Þ
where:
v D —is the drift velocity, unit m/s.
F—is the driving force of the diffusion, unit N.
k B —is Boltzmann constant.
T—is absolute temperature, K.
From Eq. (8.3) we can easily deduce that for self-diffusion, the driving force due
to concentration gradient has the following form:
F ¼ Àk T∇C
ð8:4Þ
8.2 Physics of Electromigration Process
399